A rod with length L = 1 m is exposed to a heat conduction problem where the changes of the temperature u(x) along the rod length is given by d?u du + 2- 8u = 0 dx? dx The temperatures at both ends of the rod are held constants, where at x = 0 the temperature is u(0) = 1°C and at x = 1 the temperature is u(1) = 0°C . The exact solution of the boundary value problem above is u(х) %3 (1 — еб)-1e2* + (1—е-6)-1е-4* (a) Let grid function U, U1, ..., Um,Um+1 be the approximation to the solution of u(x;), with x; = jh, h = 1/(m + 1), apply central difference scheme to approximate d²u and ", write out the (m + 2) × (m + 2) matrix A and (m + 2) vector F for du dx² dx the linear system of equations AU = F, where the boundary conditions should be [U0, U1, .., Um, Um+1]". satisfied, and the unknown U = т
A rod with length L = 1 m is exposed to a heat conduction problem where the changes of the temperature u(x) along the rod length is given by d?u du + 2- 8u = 0 dx? dx The temperatures at both ends of the rod are held constants, where at x = 0 the temperature is u(0) = 1°C and at x = 1 the temperature is u(1) = 0°C . The exact solution of the boundary value problem above is u(х) %3 (1 — еб)-1e2* + (1—е-6)-1е-4* (a) Let grid function U, U1, ..., Um,Um+1 be the approximation to the solution of u(x;), with x; = jh, h = 1/(m + 1), apply central difference scheme to approximate d²u and ", write out the (m + 2) × (m + 2) matrix A and (m + 2) vector F for du dx² dx the linear system of equations AU = F, where the boundary conditions should be [U0, U1, .., Um, Um+1]". satisfied, and the unknown U = т
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![A rod with length L = 1 m is exposed to a heat conduction problem where the changes of
the temperature u(x) along the rod length is given by
d?u
du
+ 2-
8u = 0
dx?
dx
The temperatures at both ends of the rod are held constants, where at x = 0 the
temperature is u(0) = 1°C and at x = 1 the temperature is u(1) = 0°C .
The exact solution of the boundary value problem above is
u(х) %3 (1 — еб)-1e2* + (1—е-6)-1е-4*
(a) Let grid function U, U1, ..., Um,Um+1 be the approximation to the solution of u(x;),
with x; = jh, h = 1/(m + 1), apply central difference scheme to approximate
d²u
and ", write out the (m + 2) × (m + 2) matrix A and (m + 2) vector F for
du
dx²
dx
the linear system of equations AU = F, where the boundary conditions should be
[U0, U1, .., Um, Um+1]".
satisfied, and the unknown U =
т](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F345b9968-9b02-4d06-8bef-913d0d421bb7%2F1ac503ec-a8e0-4de2-a2c5-55119b7d3cbf%2F1lkl8ig_processed.png&w=3840&q=75)
Transcribed Image Text:A rod with length L = 1 m is exposed to a heat conduction problem where the changes of
the temperature u(x) along the rod length is given by
d?u
du
+ 2-
8u = 0
dx?
dx
The temperatures at both ends of the rod are held constants, where at x = 0 the
temperature is u(0) = 1°C and at x = 1 the temperature is u(1) = 0°C .
The exact solution of the boundary value problem above is
u(х) %3 (1 — еб)-1e2* + (1—е-6)-1е-4*
(a) Let grid function U, U1, ..., Um,Um+1 be the approximation to the solution of u(x;),
with x; = jh, h = 1/(m + 1), apply central difference scheme to approximate
d²u
and ", write out the (m + 2) × (m + 2) matrix A and (m + 2) vector F for
du
dx²
dx
the linear system of equations AU = F, where the boundary conditions should be
[U0, U1, .., Um, Um+1]".
satisfied, and the unknown U =
т
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